On embedded minimal disks in convex bodies
type: Item
instance of: scholarly article
title: On embedded minimal disks in convex bodies
zbMATH Open document ID: 0617.49017
DOI: 10.1016/S0294-1449(16)30378-X
author: Michael Grüter
author: Jürgen Jost
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 1986-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/78119
review text: This paper is devoted to proving that for a bounded open strictly convex subset \(A\subset {\mathbb{R}}^ 3\) with \(C^ 4\) boundary, \(\partial A\), there exists an embedded minimal disc \(M\subset Clos(A)\) whose boundary \(\partial M\) is contained in \(\partial A\) and which is stationary (for the area integral) with respect to all variations preserving the inclusion \(\partial M\subset \partial A\), and the appropriate generalization to Riemannian manifolds is noted. The proof requires the use of methods developed in a number of different places, including the authors' own extension of W. Allard's regularity results to solutions of free boundary value problems. The argument begins by using the minimaxing methods of \textit{J. T. Pitts} [Existence and regularity of minimal surfaces on Riemannian manifolds, Math. Notes 27 (1981; Zbl 0462.58003)] to obtain a minimaxing varifold which has a certain almost minimizing property. Then the methods for minimizing among embedded surfaces of \textit{F. Almgren} and \textit{L. Simon} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 6, 447-495 (1979; Zbl 0417.49051)] and of \textit{W. Meeks}, \textit{L. Simon}, and \textit{S. T. Yau} [Ann. Math., II. Ser. 116, 621-659 (1982; Zbl 0521.53007)] are applied, together with curvature estimates extending those of \textit{R. Schoen} and \textit{L. Simon} [Commun. Pure Appl. Math. 34, 741-797 (1981; Zbl 0497.49034)] and the regularity result mentioned above, to show the almost minimizing varifold is actually an embedded minimal surface. Finally, using an argument of L. Simon and F. Smith, that embedded minimal surface is shown to be simply connected.
Mathematics Subject Classification ID: 49Q05
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 58E12
Mathematics Subject Classification ID: 49Q15
Mathematics Subject Classification ID: 49Q20
zbMATH DE Number: 4000843
zbMATH Keywords: embedded minimal disc
zbMATH Keywords: minimaxing varifold
zbMATH Keywords: curvature estimates
zbMATH Keywords: regularity
MaRDI profile type: Publication
cites work: On the first variation of a varifold
cites work: On the first variation of a varifold: Boundary behavior
cites work: Q3207672
cites work: Q5565773
cites work: Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature
cites work: The existence of minimal immersions of 2-spheres
cites work: Regularity of stable minimal hypersurfaces
cites work: Curvature estimates for minimal hypersurfaces
cites work: Stationary minimal surfaces with boundary on a simplex
cites work: On a free boundary problem for minimal surfaces
OpenAlex ID: W2541791182
description: scientific article; zbMATH DE number 4000843